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Similarity in Geometric Figures Similarity in Geometry: - The concept that shapes or other relations exist in specific ratios or proportions to one another. Ratios: When two quantities are measured in the same units, (or can be converted to the same unit of measure), they can be expressed in a constant relation to one another, referred to as a ratio. Ratios are written: a / b or a:b Simplifying Ratios: Many times, the unit of measure provided will not always be the same. In those cases they need to be converted in order to specify a meaningful ratio. 12 cm 6ft 4m 18 in. How might we convert these to simplified / meaningful ratios? 12cm 12 3 = 4* 100cm * 12 in = 18 in. 18 1

Using Ratios To Solve Problems The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. Hint: Because the ratio o AB:BC is 3:2, we can represent the length of AB as 3x and BC as 2x D C w A B l 2 l + 2w = P 2(3x) + 2(2x) = 60 6x + 4x = 60 10x = 60 X = 6 Therefore, ABCD has a length of 18 cm and a width of 12 cm

Using Ratios To Solve Problems Extended Ratios: Ratios can extend beyond a simple relationship of two measures to incorporate a third, or more, provided they are all in a constant relation to one another. In the triangle ABC, the angles exist in the following extended ratio, 1:2:3. How can we use the extended ratio to determine the measure of the angles? 2x 3x x x + 2x + 3x = 180 o Triangle Sum Theorem 6x = 180 X = 30 Therefore the angle measures are 30 o, 60 o, and 90 o

Using Ratios To Solve Problems The ratios of the side lengths of triangle DEF to the corresponding side lengths of triangle ABC are 2:1. How can we use this information to find the unknown lengths? C 3 A B F D 8 E DE is twice AB. DE = 8, so AB = ½ (8) = 4 Using the Pythagorean Theorem, we can determine BC = 5 DF is twice AC. AC = 3, so DF = 6 EF is twice BC. BC = 5, so EF = 10.

Using Proportions An equation that equates two ratios is a proportion. For example, if the ratio a/b is equal to c/d, then the following proportion can be written: a/b = c/d The numbers “a” and “d” are referred to as the extremes. The numbers “b” and “c” are referred to as the means. MeansExtremes a c b = d Properties of Proportions: 1. Cross Product Property – The product of the extremes equals the product of the means. If a/b = c/d, then ad = bc 2. Reciprocal Property – If two ratios are equal, then their reciprocals are also equal. If a/b = c/d, then b/a = d/c

Solving Proportions Using the Properties of Proportions, solve the following: 4 / x = 5 / 7 3 = 2 y + 2 y 1.Using Cross Products: 5x = 28 X = 28/5 2. Using Reciprocal Property 4/x = 5/7 x/4 = 7/5 x = 4(7/5) x = 28/5 1.Using Cross Products 3y = 2(y + 2) 3y = 2y + 4 y = 4

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Additional Properties of Geometry in Proportions Properties of Proportions: 1. Cross Product Property – The product of the extremes equals the product of the means. If a/b = c/d, then ad = bc 2. Reciprocal Property – If two ratios are equal, then their reciprocals are also equal. If a/b = c/d, then b/a = d/c 3.If a/b = c/d, then a/c = b/d 4.If a/b = c/d, then a+b/b = c+d/d

Problem Solving in Geometry with Proportions In the diagram AB / BD = AC / CE. Find the length of BD. A B C x 10 D E Given: AB _ AC BD -- CE AB _ AC BD -- CE 16/x = (30-10)/10 16/x = 20/10 20x = 160 X = 8 16/30-10 = x/10 20x = 160 X = 8 16+x/x = 30/10 30x = 10(16+x) 30x = x 20x = 160 X = 8

Geometric Mean Geometric Mean: The geometric mean of two numbers “a” and “b” is the positive number x such that: a/x = x/b. If you solve this proportion for x, through cross multiplication we find that: x 2 = a*b, therefore x = \/ a* b Example: What is the Geometric Mean of 8 and 18? 8/x = x/18  x 2 = 144  x = 12 Also, 8/12 = 12/18 because \/ 8 * 18 = \/ 144 = 12

Using Geometric Mean A4 x A3 420mm 210 mm x International Paper Standards set a standard ratio for length and width for all writing paper which is generally recognized around the world. Two paper types A3 and A4 are shown to the right. The length represented by “x” is the geometric mean of 210mm and 420mm. Find the value of x. 210 / x = x / 420 X 2 = 210 * 420 X = \/ 210 * 420 X = \/ 210 * 210 * 2 X = 210 \/ 2

Using Proportions in Real Life A Scale model of the Titanic is inches long and inches wide. The Titanic itself was feet long. How wide was it? Width of ship = x X ft / in = ft / in X = (11.25 * ) / X = 92.4 Feet

Proportions and Similar Triangles Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionately If TU || QS, then RT/TQ = RU/US > > R T U Q S Theorem 8.6 If thee parallel lines intersect two transversals, then they divide the transversals proportionately. If r|| s and s|| t, then l and m intersect r, s, t and t, then UW/WY = VX = XZ l U W Y m X Z V t r s Theorem 8.7 If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. If CD bisects <ABC, then AD/DB = CA/CB A D C B

Using Proportions in Similar Triangles In the diagram, AB||ED, BD = 8, DC = 4, and AE = 12. What is the length of EC? C 4 E D 12 8 A B Given the diagram, determine if MN || GH. G 21 M 56 H 16 N 48 L In the diagram, <1 = <2, <2 = <3. What is the length of TU? ~ ~ P S 9 11 Q T 15 R U 1 2 3